A Burnside Approach to the Termination of Mohri’s Algorithm for Polynomially Ambiguous Min-Plus-Automata Daniel Kirsten Dresden University of Technology Institut for Algebra August 31, 2006
Definition: An automaton A is called polynomially ambiguous if there exists some polynomial P : N → N such that for every w ∈ Σ ∗ there are at most P � � | w | accepting paths for w .
Definition: An automaton A is called polynomially ambiguous if there exists some polynomial P : N → N such that for every w ∈ Σ ∗ there are at most P � � | w | accepting paths for w . Theorem 1: Ibarra/Ravikumar 1986, Hromkoviˇ c/et al 2002 Let A be trim. The following assertions are equivalent: ◮ A is polynomially ambiguous. � q w � ≤ 1. ◮ For every state q , every w ∈ Σ ∗ , we have � � ❀ q ◮ For every states p , q , every w ∈ Σ ∗ , w p = q . p q = ⇒ w w w
Motivation: ◮ less explored class of automata ◮ probably a large class of feasable WFA ◮ development of proof techniques
Motivation: ◮ less explored class of automata ◮ probably a large class of feasable WFA ◮ development of proof techniques ◮ they arise in the Cauchy-product of unambiguous/ finitely ambiguous series � ( ST )( w ) := S ( u ) T ( v ) uv = w
An Example: b , 0 b , 0 1 2 3 a , 0 b , 0 a , 1 a , 0 b , 0
An Example: b , 0 b , 0 1 2 3 a , 0 b , 0 a , 1 a , 0 b , 0 � � � ◮ |A| ( w ) = min � ba ℓ b is a factor of w ℓ �
An Example: b , 0 b , 0 1 2 3 a , 0 b , 0 a , 1 a , 0 b , 0 � � � ◮ |A| ( w ) = min � ba ℓ b is a factor of w ℓ � � 1 w � ≤ | w | b − 1 < | w | . ◮ A is polynomially ambiguous, � � ❀ 3
An Example: b , 0 b , 0 1 2 3 a , 0 b , 0 a , 1 a , 0 b , 0 � � � ◮ |A| ( w ) = min � ba ℓ b is a factor of w ℓ � � 1 w � ≤ | w | b − 1 < | w | . ◮ A is polynomially ambiguous, � � ❀ 3 ◮ |A| is not the mapping of a finitely ambiguous WFA.
Mohri’s Algorithm: Let A = [ Q , θ, λ, ̺ ] be a pol. amb. WFA, i.e., ◮ Q = { 1 , . . . , n } is a finite set, ◮ θ : Σ ∗ → Z Q × Q is a homomorphism, ◮ λ, ̺ ∈ Z Q . ◮ |A| : Σ ∗ → Z , |A| ( w ) := λ θ ( w ) ̺
Mohri’s Algorithm: Let A = [ Q , θ, λ, ̺ ] be a pol. amb. WFA, i.e., ◮ Q = { 1 , . . . , n } is a finite set, ◮ θ : Σ ∗ → Z Q × Q is a homomorphism, ◮ λ, ̺ ∈ Z Q . ◮ |A| : Σ ∗ → Z , |A| ( w ) := λ θ ( w ) ̺ Let B = ( b 1 , . . . , b n ) ∈ Z Q . min( B ) := min { b i | i ∈ Q }
Mohri’s Algorithm: Let A = [ Q , θ, λ, ̺ ] be a pol. amb. WFA, i.e., ◮ Q = { 1 , . . . , n } is a finite set, ◮ θ : Σ ∗ → Z Q × Q is a homomorphism, ◮ λ, ̺ ∈ Z Q . ◮ |A| : Σ ∗ → Z , |A| ( w ) := λ θ ( w ) ̺ Let B = ( b 1 , . . . , b n ) ∈ Z Q . min( B ) := min { b i | i ∈ Q } � � nf( B ) := ( − min( B )) + B = b 1 − min( B ) , . . . , b n − min( B )
Mohri’s Algorithm: Let A = [ Q , θ, λ, ̺ ] be a pol. amb. WFA, i.e., ◮ Q = { 1 , . . . , n } is a finite set, ◮ θ : Σ ∗ → Z Q × Q is a homomorphism, ◮ λ, ̺ ∈ Z Q . ◮ |A| : Σ ∗ → Z , |A| ( w ) := λ θ ( w ) ̺ Let B = ( b 1 , . . . , b n ) ∈ Z Q . min( B ) := min { b i | i ∈ Q } � � nf( B ) := ( − min( B )) + B = b 1 − min( B ) , . . . , b n − min( B ) nf((1 , 2 , 3)) = (0 , 1 , 2) nf((3 , ∞ , 4)) = (0 , ∞ , 1) nf((3 , ∞ , − 4)) = (7 , ∞ , 0)
Let Q ′ ⊆ Z Q be the least set which satisfies ◮ nf( λ ) ∈ Q ′ , and
Let Q ′ ⊆ Z Q be the least set which satisfies ◮ nf( λ ) ∈ Q ′ , and ◮ for every B ∈ Q ′ , a ∈ Σ, � � ∈ Q ′ . nf B θ ( a )
Let Q ′ ⊆ Z Q be the least set which satisfies ◮ nf( λ ) ∈ Q ′ , and ◮ for every B ∈ Q ′ , a ∈ Σ, � � ∈ Q ′ . nf B θ ( a ) We have Q ′ = � w ∈ Σ ∗ � � � nf( λθ ( w )) .
Let Q ′ ⊆ Z Q be the least set which satisfies ◮ nf( λ ) ∈ Q ′ , and ◮ for every B ∈ Q ′ , a ∈ Σ, � � ∈ Q ′ . nf B θ ( a ) We have Q ′ = � w ∈ Σ ∗ � � � nf( λθ ( w )) . Mohri’s Algorithm uses the set Q ′ as states. It terminates iff Q ′ is finite.
An Example: w , 3 1 2 w , 2 w , 1
An Example: w , 3 1 2 w , 2 w , 1 For k ≥ 1, we have λ ( θ ( w )) k = (2 k , k ) and
An Example: w , 3 1 2 w , 2 w , 1 For k ≥ 1, we have λ ( θ ( w )) k = (2 k , k ) and � λ ( θ ( w )) k � nf = ( k , 0), i.e., Mohri’s algorithm does not terminate on the sequence ( w k ) k ≥ 1 .
Another Example: w , 3 1 2 w , 1 w , 2
Another Example: w , 3 1 2 w , 1 w , 2 For k ≥ 2, we have λ ( θ ( w )) k = ( k , k + 2) and
Another Example: w , 3 1 2 w , 1 w , 2 For k ≥ 2, we have λ ( θ ( w )) k = ( k , k + 2) and � λ ( θ ( w )) k � nf = (0 , 2), i.e., Mohri’s algorithm terminates on the sequence ( w k ) k ≥ 1 .
Let w ∈ Σ ∗ and B = θ ( w ). Assume that B has an idempotent structure, i.e., � � B [ i , j ] � = ∞ ⇐ ⇒ BB [ i , j ] � = ∞ for all i , j ∈ Q .
Let w ∈ Σ ∗ and B = θ ( w ). Assume that B has an idempotent structure, i.e., � � B [ i , j ] � = ∞ ⇐ ⇒ BB [ i , j ] � = ∞ for all i , j ∈ Q . For i , j ∈ Q let i ≤ B j iff B [ i , j ] � = ∞ .
Let w ∈ Σ ∗ and B = θ ( w ). Assume that B has an idempotent structure, i.e., � � B [ i , j ] � = ∞ ⇐ ⇒ BB [ i , j ] � = ∞ for all i , j ∈ Q . For i , j ∈ Q let i ≤ B j iff B [ i , j ] � = ∞ . The relation ≤ B is transitive and antisymmetric,
Let w ∈ Σ ∗ and B = θ ( w ). Assume that B has an idempotent structure, i.e., � � B [ i , j ] � = ∞ ⇐ ⇒ BB [ i , j ] � = ∞ for all i , j ∈ Q . For i , j ∈ Q let i ≤ B j iff B [ i , j ] � = ∞ . The relation ≤ B is transitive and antisymmetric, but not necessarily reflexive of irreflexive, i.e., ≤ B is almost a partial ordering.
A subset C ⊆ Q is a clone iff there exists some v ∈ Σ ∗ such that C = � λθ ( v )[ i ] � = ∞ � � � i ∈ Q .
A subset C ⊆ Q is a clone iff there exists some v ∈ Σ ∗ such that C = � λθ ( v )[ i ] � = ∞ � � � i ∈ Q . � B [ i , j ] � = ∞ for some i ∈ C � � � C and B are stable iff C = j ∈ Q .
A subset C ⊆ Q is a clone iff there exists some v ∈ Σ ∗ such that C = � λθ ( v )[ i ] � = ∞ � � � i ∈ Q . � B [ i , j ] � = ∞ for some i ∈ C � � � C and B are stable iff C = j ∈ Q . C and B satisfy the clones property if for every i ∈ C which is minimal w.r.t. ≤ B , the value B [ i , i ] is minimal among B [ j , j ] for j ∈ C .
A subset C ⊆ Q is a clone iff there exists some v ∈ Σ ∗ such that C = � λθ ( v )[ i ] � = ∞ � � � i ∈ Q . � B [ i , j ] � = ∞ for some i ∈ C � � � C and B are stable iff C = j ∈ Q . C and B satisfy the clones property if for every i ∈ C which is minimal w.r.t. ≤ B , the value B [ i , i ] is minimal among B [ j , j ] for j ∈ C . Lemma: � k ∈ N � nf( λθ ( vw k )) � � The set is finite iff C and B satisfy the clones property.
A pol. amb. WFA A satisfies the clones property if ◮ for every clone C ,
A pol. amb. WFA A satisfies the clones property if ◮ for every clone C , ◮ for every w ∈ Σ ∗ such that B := θ ( w ) has an idempotent structure and C and B are stable,
A pol. amb. WFA A satisfies the clones property if ◮ for every clone C , ◮ for every w ∈ Σ ∗ such that B := θ ( w ) has an idempotent structure and C and B are stable, C and B satisfy the clones property.
A pol. amb. WFA A satisfies the clones property if ◮ for every clone C , ◮ for every w ∈ Σ ∗ such that B := θ ( w ) has an idempotent structure and C and B are stable, C and B satisfy the clones property. Kirsten 2005 Theorem 2: Let A be trim, polynomially ambiguous WFA. The following assertions are equivalent: 1. Mohri’s algorithm terminates on A .
A pol. amb. WFA A satisfies the clones property if ◮ for every clone C , ◮ for every w ∈ Σ ∗ such that B := θ ( w ) has an idempotent structure and C and B are stable, C and B satisfy the clones property. Kirsten 2005 Theorem 2: Let A be trim, polynomially ambiguous WFA. The following assertions are equivalent: 1. Mohri’s algorithm terminates on A . 2. For every v , w ∈ Σ ∗ , Mohri’s algorithm terminates on the sequence ( vw k ) k ≥ 1 on A . 3. The WFA A satisfies the clones property.
A bad Example: b , 0 1 2 3 a , 1 a , 0 b , 0 a , 0 b , 1
A bad Example: b , 1 b , 0 1 2 3 a , 0 a , 1 a , 0 b , 0 a , 0 b , 1
A bad Example: b , 1 b , 0 1 2 3 a , 0 a , 1 a , 0 b , 0 a , 0 b , 1 ◮ For every v , w ∈ Σ ∗ , Mohri’s algorithm terminates on ( vw k ) k ≥ 1 .
A bad Example: b , 1 b , 0 1 2 3 a , 0 a , 1 a , 0 b , 0 a , 0 b , 1 ◮ For every v , w ∈ Σ ∗ , Mohri’s algorithm terminates on ( vw k ) k ≥ 1 . ◮ Mohri’s algorithm does not terminate on baba 2 ba 3 ba 4 b . . . (2) ⇒ (1) in Theorem 2 does not hold for A .
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